Coordinate Geometry
Circles
MMTS_Full_Test_09
Grade 12
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 $\left(-\frac{1}{7},3\right)$ intersect the circle at only one point $p(\alpha,\beta)$. Then $\dfrac{29}{6}(\beta-\alpha)$ is equal to
Step-by-Step Solution
Key Concept: Centre = intersection of diameters; the line is tangent at $p$
Centre: $2x-3y=5$ and $3x-4y=7$: $x=1,y=-1$. The given line's equation: through $(-22/7,-4)$ and $(-1/7,3)$: slope$=\frac{7}{3}$. Line touches circle at $p$: $\frac{29}{6}(\beta-\alpha)=7$.
Correct Answer: 7