Circles
Chord of a Circle / Intersection Point
nta_pyq_2024_jan
Grade 11
Question:
Equations of two diameters of a circle are $2x-3y=5$ and $3x-4y=7$. The line joining the points $\left(-\frac{22}{7},-4\right)$ and $\left(-\frac{1}{7},3\right)$ intersects the circle at only one point $P(\alpha,\beta)$. Then $17\beta-\alpha$ is equal to
Step-by-Step Solution
Key Concept: Centre of circle = intersection of diameters $2x-3y=5$ and $3x-4y=7$: solving gives $(1,-1)$. Find equation of line $AB$ through the two given points. The line is tangent/intersects at one point; find $P$ using perpendicular from centre.
Line $AB$: $7x-3y+10=0$. Perpendicular from $C(1,-1)$: $3x+7y+4=0$. Solving: $\alpha=-41/29$, $\beta=1/29$. $17\beta-\alpha=17/29+41/29=58/29=2$.
Correct Answer: 2