JEE Mathematics Question

Given: The line is x + y = 7 and the point (, 7 33) lies on the curve traced by the mid-points of the line segments between the coordinate axes. Find: and hence the correct option. The solution states that the intercepts of the line are A ( 7, 0), B (0, 7) Therefore, the midpoint M(h,k) is h = 72, k = 72 Using the given ordinate, k = 7 33 Substitute into k = 72 to get 72 = 7 33 Hence, = 32 so = 3 Now use h = 72 Therefore, = 72 ( 3) = 72 12 = 7 Thus the computed value is 7. However, the solution explicitly marks The Correct Option is C, which does not match the computed value or the listed options. the answer is recorded as C despite this discrepancy.

JEE Question — Solution | JEE Mathematics

MCQMediumJEE 2023Conic Sections (Parabola, Ellipse, Hyperbola)

Question image showing a coordinate geometry problem about the locus of midpoints of line segments cut by the line x cos theta + y sin theta = 7 between coordinate axes, asking for alpha when point (alpha, 7√3/3) lies on the curve.

A
$7$
B
$-7$
C
$-7\sqrt{3}$
D
$7\sqrt{3}$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: The line is

x\cos\theta + y\sin\theta = 7

and the point $\left(\alpha, \frac{7\sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments between the coordinate axes.

Find: $\alpha$ and hence the correct option.

The solution states that the intercepts of the line are

A\left(\frac{7}{\cos\theta}, 0\right), \quad B\left(0, \frac{7}{\sin\theta}\right)

Therefore, the midpoint $M(h,k)$ is

h = \frac{7}{2\cos\theta}, \quad k = \frac{7}{2\sin\theta}

Using the given ordinate,

k = \frac{7\sqrt{3}}{3}

Substitute into

k = \frac{7}{2\sin\theta}

to get

\frac{7}{2\sin\theta} = \frac{7\sqrt{3}}{3}

Hence,

\sin\theta = \frac{\sqrt{3}}{2}

\theta = \frac{\pi}{3}

Now use

h = \frac{7}{2\cos\theta}

Therefore,

\alpha = \frac{7}{2\cos\left(\frac{\pi}{3}\right)} = \frac{7}{2\cdot \frac{1}{2}} = 7

Thus the computed value is $7$ . However, the solution explicitly marks The Correct Option is C, which does not match the computed value or the listed options. the answer is recorded as C despite this discrepancy.

Common mistakes

Using the intercepts as $\left(7\cos\theta,0\right)$ and $\left(0,7\sin\theta\right)$ is wrong because intercepts are found by setting one coordinate to zero and solving. The correct intercepts are $\left(\frac{7}{\cos\theta},0\right)$ and $\left(0,\frac{7}{\sin\theta}\right)$ .
Forgetting the midpoint formula leads to missing the factor of $2$ in the denominator. Since the midpoint averages coordinates, use $h = \frac{7}{2\cos\theta}$ and $k = \frac{7}{2\sin\theta}$ .
After obtaining $\sin\theta = \frac{\sqrt{3}}{2}$ , taking an incompatible value of $\theta$ ignores the given interval $\theta \in \left(0, \frac{\pi}{2}\right)$ . The correct choice is $\theta = \frac{\pi}{3}$ within that interval.

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Answer

Step-by-step solution

Common mistakes

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